Have you checked the event log on the application server to see if there is anything obvious in there? If not, I would increase the logging level on the inbound mail service so you can see what is actually happening when it tries to process the mailbox.

In Configuration Center, stop the inbound mail service and click edit. Change the Log Severity from Error to Trace. Click ok and then start the service. You should then see what the mail service is doing in the event log.